Main

The phase of an electromagnetic wave is shifted on scattering from an atom, a microscopic effect that adds up coherently and determines the macroscopic properties of wave propagation, as described by the index of refraction n. Near atomic transitions, resonant effects become particularly pronounced and a wealth of information about photon–atom interactions and the involved atomic energy levels is encoded in the optical response. For both resonant and non-resonant cases, wave propagation phenomena are relevant for nearly all methods for the manipulation of light. Examples are plenty: from passive shaping with lenses and phase plates, guiding of light in fibre optics and waveguides, all the way to the precise active control, for example, in liquid crystal displays, electro-optical modulators and integrated photonic devices. These optical properties and functionalities crucially rely on precisely known and controllable phase shifts. At the same time, when a wave propagates through an object with a spatially dependent index of refraction n(r), the local variations in the accumulated phase shifts reflect its internal structure, offering a contrast mechanism for phase-sensitive imaging. Note that phase contrast can not only be substantially more prominent than attenuation contrast1 (for example, in X-ray imaging of biological matter) but also more dose efficient2, because the underlying elastic scattering process does not deposit any energy in the specimen.

Guided modes offer a particular benefit to control light propagation and phase effects. In optical waveguides, photons can be coupled to individual quantum emitters such as resonant atoms, in a controllable manner. This is exploited in the emerging field of waveguide quantum electrodynamics to investigate quantum many-body physics and to develop integrated functional quantum technologies3,4. The resulting quantum-optical systems are often nonlinear3 and can become increasingly complex such that refractive index models are no longer valid. Yet, the phase shift imposed by individual emitters remains a central quantity, which can even become a function of the number of photons5. Further, the phase shift can be manipulated, for example, in quantum phase switches with single atoms6,7, a building block that may pave the way to photon–photon quantum gates. To observe any of this experimentally, one crucially relies on methods for the precise and accurate measurements of phase shifts.

In the optical regime, instruments such as the Mach–Zehnder interferometer are readily available to measure phase shifts. They can be assembled on optical tables from discrete components, but are also routinely realized in integrated photonic devices. In the X-ray regime, on the other hand, the situation is fairly different, because the short wavelengths require extreme stability and translation precision in the order of picometres. Bonse and Hart realized the first interferometer for hard X-rays8, a Mach–Zehnder-type setup, achieving the necessary stability by cutting the entire interferometer from one solid single crystal. This principle, based on crystal reflections9,10,11, has enabled the design of interferometers that have been used to measure refractive indices and dispersion corrections to the atomic form factors close to absorption edges12,13. In most cases, however, refractive indices in the hard X-ray regime are obtained indirectly from absorption spectra by employing Kramers–Kronig relations14,15, a method that relies on precisely measured absorption spectra over a wide frequency range. Unfortunately, neither bulky crystal interferometers nor any of the other methods15 are suitable to measure phase shifts of nanometre-sized beams or in integrated geometries.

This limitation is particularly regrettable, since X-ray photon–matter interaction within nanostructures exhibits many interesting phenomena16. These include X-ray generation in nanostructures17, Purcell-enhanced X-ray scintillation and enhanced interaction in waveguide cavities18. Nanometre-sized X-ray waveguides have been realized for double-waveguide phase-contrast imaging19, curved X-ray waveguides20 and integrated X-ray beamsplitters21, representing the first steps towards ‘X-ray optics on a chip’. Recently, ensembles of Mössbauer nuclei coupled to an X-ray waveguide were experimentally investigated with synchrotron radiation, demonstrating a platform for waveguide quantum electrodynamics in the hard X-ray regime22,23, representing the first step towards the realization of more advanced theoretical concepts24,25. Such Mössbauer isotopes constitute exceptionally clean quantum systems due to their extremely narrow resonances26, making them a particularly successful experimental platform for quantum optics in the hard X-ray frequency regime27,28,29,30,31,32. Despite its tremendous importance throughout these examples of photon–matter interaction within nanostructures, phase shifts have been elusive to measurements in such geometries.

In this work, we interferometrically measure the phase shift that an ultrathin layer of 57Fe Mössbauer nuclei coherently imprints onto photons propagating through a single-mode X-ray waveguide. To that end, we have devised a nanoscale double-waveguide interferometer, reminiscent of Young’s celebrated double-slit experiment33. It coherently superimposes the exit fields of the signal waveguide (SWG) and a reference waveguide (RWG) free of resonant nuclei, and converts the resonant phase shift in the SWG into a measurable lateral shift of the far-field interference pattern (Fig. 1b). This allows to extract the nuclear phase shift from just a small number of detected photons. The isotope 57Fe has a low-lying nuclear resonance at ℏω0 = 14.41 keV with a linewidth of only ℏΓ = 4.7 neV (Fig. 1a). For the nuclear isomers of Mössbauer isotopes with their particular timescales and energy scales, specialized methods to determine the phase shifts of resonantly scattered synchrotron radiation have been proposed34 and demonstrated35,36,37,38,39,40,41. These bear certain analogies to in-line interferometers in which the sample and a reference sample with shifted resonance energy are behind one another in the same optical path. The two resonant samples are illuminated with short broadband pulses and their combined time-domain response functions are recorded. To extract the spectral phase shift, these methods rely on computational reconstruction from the measured temporal signatures, which requires a large number of detected photons41. By contrast, here we measure the spectral phase shift ϕ(Δ) resolved in detuning from the resonance Δ = ω − ω0 of the light frequency ω to resonance ω0 using quasi-monochromatic narrowband illumination by coherently interfering individual photons from two spatially separated beams. The measured phase shifts unambigously reveal the value of hyperfine broadening and the coupling strength between the nuclei and photons propagating in the waveguide, beyond the information that is accessible from absorption measurements alone.

Fig. 1: Principle of a double-waveguide interferometer.
Fig. 1: Principle of a double-waveguide interferometer.
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a, A resonant atom (here the nuclear resonance of 57Fe between ground and excited states with nuclear spin Ig = 1/2 and Ie = 3/2, respectively) causes a phase shift depending on the detuning Δ = ω − ω0 of the light frequency ω to resonance ω0. This is described by the real part of the complex susceptibility χ. b, Ensemble of resonant atoms (red) in a nanoscopic waveguide coherently shifts the phase of photons propagating through this SWG compared with an RWG without the atoms. c, Phase shift results in a shift of the far-field double-slit interference pattern (filled blue curve) of the two exit waves compared with the unshifted pattern (dashed line). Blocking the RWG results in a single-slit diffraction pattern (green filled curve) that is unaffected by the phase shift.

Results

To implement the nano-interferometer, we fabricated a pair of 0.5-mm-long planar X-ray waveguides, stacked on top of each other, one of which (the SWG) contains a 0.6-nm-thick film (about two atomic layers) of isotope-enriched 57Fe (Fig. 1). The waveguides consist of a 20-nm-thick boron carbide (B4C) guiding core surrounded by 30 nm of molybdenum cladding, resulting in the interferometer baseline of 50 nm. The design ensures that only a single guided mode of the electromagnetic field is transmitted by each waveguide. The waveguides are enclosed in 1-mm-thick germanium wafers to block the tails of the illuminating focus42. In addition, as control experiments, we have prepared a double waveguide without the resonant nuclei, as well as a single waveguide containing resonant nuclei. The latter conceptually corresponds to blocking the reference ‘slit’.

The experiment requires a beam of hard X-rays at photon energy ℏω0 = 14.4 keV, where ℏ is the reduced Planck’s constant, which is both highly monochromatic, on the order of the natural linewidth of 57Fe, and tightly focused, to couple efficiently into the front face of the two single-mode waveguides43. To this end, the experiments were performed at the nuclear resonance beamline ID18 (ref. 44) of the European Synchrotron Radiation Facility (ESRF), utilizing the Synchrotron Mössbauer Source (SMS)45 to provide a monochromatic beam, which is subsequently focused to a spot of about 0.5 μm in diameter by a pair of elliptical focusing mirrors in the Kirkpatrick–Baez geometry46. A sketch of the experimental setup is shown in Fig. 2. The SMS uses an electronically forbidden but nuclear-allowed Bragg reflection of an isotopically enriched iron borate (57FeBO3) crystal to extract a bandwidth of a few nanoelectronvolts with practically zero background47. To tune the photon energy over the target resonance, iron borate is mounted on a Doppler velocity transducer, which is harmonically oscillating (the motion is harmonic for optimal stability), thereby periodically shifting the energy via the Doppler effect. The photons transmitted by the waveguides are detected by a time-resolving pixel array detector based on Timepix3 (ref. 48). The detector is synchronized to the velocity transducer such that the photon time of arrival encodes the transducer oscillation phase and, consequently, the detuning of the photon energy. The detector is installed 1 m downstream of the waveguides, where the 55-μm pixel pitch provides an angular resolution of the interference pattern of about 0.003°, with a total photon detection rate on the order of 10 s−1.

Fig. 2: Experimental scheme of the waveguide nano-interferometer.
Fig. 2: Experimental scheme of the waveguide nano-interferometer.
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a, Broadband undulator radiation, pre-monochromatized to about 2 eV with a high-heat-load monochromator (HHLM), is further monochromatized to the nuclear linewidth by the SMS, then focused by elliptical mirrors in the Kirkpatrick–Baez (KB) geometry and front-coupled into the double waveguide (WG). The far-field pattern of the double WG is observed by a time-resolving pixel detector (Timepix3). Internally, the SMS consists of a high-resolution monochromator (HRM), a deflector and an iron borate (IB) crystal. The IB is mounted on a velocity transducer and harmonically oscillating with velocity v(t), so that the photon energy is periodically Doppler shifted by Δ(t). The oscillation is controlled by a function generator (FG). The detector is synchronized to the IB oscillation such that the photon time of arrival encodes the instantaneous frequency detuning. b, Cross-sectional scanning transmission electron microscopy image of the double WG.

Figure 3 shows the measured far-field interference patterns as a function of photon energy detuning (Methods provides a detailed account of the data preprocessing) for the three interferometer configurations: with resonant nuclei in the SWG (Fig. 3a), with nuclei in the SWG but without reference (analogous to a ‘blocked reference slit’; Fig. 3b), and both waveguides without resonant nuclei (Fig. 3c). Hence, Fig. 3b,c can be considered ‘control’ experiments to the case of interest, namely, Fig. 3a. Figure 3d–f shows the corresponding absorption spectra, obtained by summing the interference pattern column wise. The single waveguide in Fig. 3e exhibits a broadened resonant absorption dip, which is reproduced by the interferometer in Fig. 3d with a unit offset due to the RWG. Note that the interference patterns encode the phase difference between the fields at the exits of the two waveguides, including shifts accumulated within the waveguides as well as phase differences already present at the waveguide entrances. In particular, a slightly tilted wave front results in a frequency-independent shift of the interference pattern, a global offset. This effectively limited the maximum acquisition time and, consequently, the total number of detected photons in the interference patterns, because of gradual drifts in the wave front of the illuminating focus, probably due to the ambient temperature (Methods). This reduces the fringe visibility over time.

Fig. 3: Measured energy-resolved interference patterns and the corresponding absorption spectra.
Fig. 3: Measured energy-resolved interference patterns and the corresponding absorption spectra.
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Three different cases are illustrated, within which the red line indicates the layer of Mössbauer nuclei. a–c, Far-field interference pattern in terms of exit angle ξ and detuning Δ = ω − ω0 for the three cases: SWG and RWG (a), SWG only (b) and two RWGs (no Mössbauer nuclei; c). The colour encodes the total number of detected photons per bin. d–f, Waveguide transmission coefficients (Mössbauer absorption spectra) for the three cases (a–c), respectively (column-wise sum of the interference patterns). I0 indicates the total number of detected photons per interference pattern.

The far-field interference pattern can be modelled by (Methods)

$$I(\xi )={I}_{0}(\xi )\left[1+T+{V}_{0}\sqrt{T}\cos (\phi +kb\xi )\right],$$
(1)

where I0(ξ) is the far-field diffraction pattern of a single waveguide mode, ξ is the observation angle, b is the baseline (waveguide separation in z), ϕ is the phase shift between the two waveguides, k = ω0/c is the on-resonance wavenumber and T is the intensity transmission coefficient of the SWG. The parameter V0 ≤ 1 accounts for a global loss of fringe visibility. On resonance, the nuclei fully absorb the photons (T = 0), which results in 0% and 50% transmission in the single and double waveguide, respectively. Further away from resonance, the interference pattern is clearly visible and continuously shifts with varying detuning.

From the measured interference pattern, we have extracted the phase shifts ϕ(Δ) induced by the nuclear resonance, using Bayesian inference49 taking the Poissonian probability distribution of the data into account. To that end, we have computed the full posterior probability density of the phase shift p(ϕ∣data), conditional on the respective observed interference pattern, using a Markov chain Monte Carlo ensemble sampler50, independently for each frequency detuning Δ. This approach gives us not only the most likely value of the phase shift but its entire probability distribution and, hence, a robust estimate of uncertainty.

Figure 4 shows the resulting probability densities for the phase shift. For larger detunings, the phase shift is precisely localized (standard deviation of 0.13 rad) and exhibits the expected resonance. On the other hand, within ±6Γ of resonance, the visibility is practically zero (compare Fig. 3a) such that the interference pattern contains no information on the phase shift. Note that this is not specific to the method but a general consequence of the practically vanishing transmission. As shown by the values of I0 in Fig. 3d, the phases were extracted from interference patterns with fewer than 300 photons each.

Fig. 4: Extracted phase shifts.
Fig. 4: Extracted phase shifts.
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Phase shift ϕ extracted from the data shown in Fig. 3a by the Bayesian phase retrieval procedure. The colour encodes the posterior probability density for ϕ(Δ) given the measured interference patterns at a certain detuning Δ. Due to strong absorption, the phase is undetermined in a frequency band of ±6Γ around the resonance frequency. Outside of that band, the phase is well determined.

Connecting the phase shifts to photon–matter interaction

The phase shift contains information about the photon–nuclei coupling and helps to disambiguate parameters that lead to similar absorption spectra. On the basis of a recently developed theoretical model22, the harmonic components of the field propagating (along x) through the waveguide and coherently interacting with the nuclei can be written as

$$E(\omega ,x)={E}_{0}(\omega ,x)\exp \left[{\rm{i}}k{\zeta }_{{\rm{m}}}\chi (\omega )x/2\right],$$
(2)

where E0 is the reference field without nuclei and ζm quantifies the coupling between the single guided mode and the nuclear layer (Methods). The bare nuclear resonance corresponds to the magnetic susceptibility

$${\chi }_{0}(\omega )=-{\rho }_{{\rm{n}}}\frac{{\sigma }_{\mathrm{rad}}}{k}\frac{\varGamma /2}{\omega -{\omega }_{0}+{\rm{i}}\varGamma /2},$$
(3)

where σrad is the radiative cross-section, k = ω0/c is the wavenumber and ρn is the number density of resonant nuclei (Fig. 1a). The nuclear levels are generally split due to electric and magnetic hyperfine interactions, leading to a modified susceptibility χ. Since the thin iron layer (which is only about two atomic layers thick) does not develop long-range magnetic order at room temperature, we do not expect magnetic hyperfine splitting here. Yet, even small differences in the environment and electric field gradients of the nuclei in the thin iron film lead to inhomogeneous line broadening and possibly a quadrupole splitting of the resonance line. We assume a Gaussian-broadened susceptibility χ governed by two effective hyperfine parameters—net central shift and line broadening (Methods).

Figure 5a shows the measured absorption spectrum and two theory curves based on equation (2), for which the hyperfine parameters and coupling strength \(\left|{\zeta }_{{\rm{m}}}\right|\) were found with maximum likelihood estimations with respect to either the absorption spectrum alone (dashed line) or the absorption spectrum combined with the phase shifts (solid line). More precisely, the former shows the model for which the measured absorption spectrum is the most likely, with no regard for the phase shift. For the latter, on the other hand, the two hyperfine parameters were optimized with respect to the absorption spectrum and the coupling strength with respect to the phase shift. In both cases, only the modulus \(\left|{\zeta }_{{\rm{m}}}\right|\) of the complex coupling coefficient \({\zeta }_{{\rm{m}}}=\left|{\zeta }_{{\rm{m}}}\right|\exp [{\rm{i}}\arg {\zeta }_{{\rm{m}}}]\) was varied and the complex argument was fixed at the value \(\arg {\zeta }_{\mathrm{m}}^{\dagger }=0.03\), which was obtained from simulations using the design parameters of the layer structure.

Fig. 5: Absorption spectra and phase shifts.
Fig. 5: Absorption spectra and phase shifts.
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a, Measured intensity transmission (blue diamonds) as a function of detuning Δ = ω − ω0 together with two model curves (see the main text). b, Extracted phase shifts (red dots; first moment of the posterior probability density) together with the corresponding model curves from a. The red area indicates the ±6Γ band of nearly full absorption, for which the data does not contain any phase information. The same model was optimized with respect to either the absorption spectrum alone (dashed grey line) or the absorption spectrum combined with the phase shifts (solid black line).

The parameters extracted in this way are listed in Table 1. The coupling strength of the combined model closely matches the design value from the theory calculations of \(\left|{\zeta }_{\mathrm{m}}^{\dagger }\right|=7.1\times 1{0}^{-3}\) (Methods), whereas the absorption model corresponds to a substantially smaller coupling of 4.6 × 10−3. Both models describe the absorption spectrum (Fig. 5a) well, which is the result of line broadening and the coupling strength being anticorrelated with respect to the absorption spectra. In the phase shifts (Fig. 5b), however, we can clearly disambiguate the two models. Although the first model does not seem to reproduce the observed phase shifts, the second model does. Thus, only the combination of the two observables allows us to uniquely extract the model parameters, particularly quantifying the coupling parameter.

Table 1 Hyperfine parameters and coupling strength extracted from the data
Full size table

The higher sensitivity and information content of the phase compared with the absorption signal can be attributed to the fact that the phase shift asymptotically decays like ϕ ∝ Δ−1, whereas the absorbance decays like μ ∝ Δ−2, where μ is the absorbance. Consequently, even noisy measurements of the phase shift reveal quantitative information over a frequency range much wider than the width of the absorption line. This is clear evidence that Mössbauer spectroscopy, in general, can distinctly benefit from the phase information.

Discussion

In summary, we have measured the resonant phase shift imprinted onto photons that propagate through an X-ray waveguide, and coherently interacting with an ensemble of nuclear resonances, using a nano-interferometer operated in the single-photon regime. The method is based on two-beam interference of individual monochromatic photons. This has allowed us to accurately extract the phase shift from only a few 100 detected photons, independently for each sampled photon energy. By using Bayesian inference, we retrieve the full probability distributions of the phase shifts, in contrast to previously demonstrated methods. Simulations suggest that accuracy and uncertainty improve with \(1/\sqrt{N}\) with increasing counts N (see Extended Data Figs. 1 and 2 and Supplementary Section 1). Thus, this method could resolve much weaker phase shifts than demonstrated here. Using the extracted phase shift, we were able to quantify the coupling strength between the waveguide mode and the nuclei, which was not possible from the absorption spectrum alone, since line broadening and optical thickness lead to ambiguous signatures (Fig. 5). The observed phase shift and absorption spectra are consistently described by a recently derived theoretical model22,51.

Our results demonstrate the interferometric measurement of phase shifts in the X-ray regime on the nanometre scale, using a few individual photons. The technique is not restricted to the 57Fe nuclear resonance but can analogously be applied to other nuclear and electronic resonances, as X-ray waveguides are well established for photon energies in the range of 6 keV to 20 keV, but not limited to this range. This paves the way for a new generation of X-ray photonics on a chip for synchrotron radiation and potentially laboratory sources (Supplementary Section 2), combining curved waveguides20, beamsplitters21 and integrated X-ray sources17 with integrated interferometric sensors.

Having accurately measured the phase shift, one can envision its control. A thicker layer of resonant iron nuclei will develop a long-range magnetic order such that the degenerate resonance splits into a sextet. The phase shift for a fixed photon energy could then be tuned by controlling the magnetic hyperfine spectrum52,53, either by applying a static magnetic field or transiently with radio-frequency pulses30. This would enable the active control of the phase for narrowband pulses in a nanophotonic device in the hard X-ray frequency range.

Methods

Waveguide design

The waveguides were designed using the xwglib software51. Materials, layer thicknesses and length are a trade-off between effectively absorbing everything but a single guided mode of the electromagnetic field and maximizing the transmission of that mode. The simulated attenuation length (intensity of 1/e) of the two supported guided modes are 0.43 mm and 0.01 mm (0.31 and 4 × 10−3 transmission over 0.5 mm, respectively). Thus, the waveguide essentially supports only a single mode, as the second mode is strongly attenuated with respect to the first. The outer germanium attenuates the non-guided field by 10−12. The iron layer was dimensioned to produce a resonant phase shift in the order of ±π. Spacing between the waveguides, b = 50 nm, was chosen to be large enough to ensure decoupled modes and still resolve the interference fringes on the detector ~1 m downstream.

Waveguide fabrication

The waveguide samples were fabricated via magnetron sputter deposition, using an argon plasma at a pressure of 5 × 10−3 mbar at a target-to-substrate distance of about 7 cm with sputtering guns from the A300-XP series (AJA International). The residual base pressure before deposition was typically less than 4 × 10−7 mbar. Molybdenum and the 57Fe-isotope-enriched iron (97%) were deposited using targets with a diameter of 3.81 cm at 12-W d.c., and B4C using a target with diameter of 5.08 cm at 53-W radio-frequency power. The deposition rates were calibrated based on X-ray reflectivity measurements. The waveguides were deposited onto 1-mm-thick polished germanium wafers. To block the over-illuminating part of the X-ray focus, a second 1-mm-thick germanium wafer was bonded on top of the layer structure, using soft-alloy bonding42.

High-angle annular dark-field scanning transmission electron microscopy imaging was performed in an FEI Titan 80-300 E-TEM device using an acceleration voltage of 300 kV and a convergence semiangle of 10 mrad, yielding a probe size of approximately 0.14 nm.

Event-based detection

To detect photons in the far field, we used an AdvaPIX TPX3 (ADVACAM) device with a 500-μm-thick silicon sensor (70% efficiency at 14.4-keV photon energy). We operated this Timepix3-based48,54 detector in the event-based mode, such that it produces a list of detection events with pixel coordinates, time of arrival (ToA) and time-over-threshold data. The velocity transducer of the SMS was controlled by a DFG-1200 digital function generator (Wissenschaftliche Elektronik) set to the sinusoidal mode. We fed the transistor–transistor logic output of the DFG through a custom-built signal converter into the external synchronization interface of the AdvaPIX. In this way, we set the detector to reset its ToA counter in each oscillation period of the velocity transducer. The temporal resolution of the synchronization was limited by the detector clock frequency of 40 MHz—significantly exceeding the resolution required for the 10-Hz oscillation of the velocity transducer. As a result, the ToA data of the detected photons encode the velocity of the transducer and, therefore, the energy detuning.

The total number of detected photons is limited by the fact that the interference pattern is highly sensitive to small relative changes in the wave front (phase) of the illuminating focus. It could only be kept stable for about 1 h because the beamline optics had to be periodically realigned due to temperature drifts in the building, which slightly altered the wave front in focus and shifted the interference patterns. We remark that this instability has been substantially reduced shortly after our experiment, as the beamline has been moved into temperature-controlled hutches.

Photon energy calibration

Photon energy detuning was calibrated based on the absorption spectrum of a natural α-iron reference absorber foil used for synchrotron and conventional Mössbauer spectroscopy. Zero detuning (Δ = 0) corresponds to the centre of the α-iron absorption spectrum.

Data preprocessing

Several preprocessing steps were applied to the data to generate the interference patterns.

  1. 1.

    Apply a photon energy filter to reduce spurious events due to environmental and cosmic radiation as well as charge-sharing events. For this, events outside a time-over-threshold range were discarded.

  2. 2.

    Select data from a time window during which the beam was the most stable by visual inspection.

  3. 3.

    Select a horizontal range of the data that is as homogeneous as possible by visual inspection. This was necessary because the illuminating wave field had a visibly tilted phase gradient.

  4. 4.

    Project the detection position onto the vertical axis.

  5. 5.

    Convert ToA data into energy detuning based on the calibration and vertical pixel position z to raw observation angle, that is, \({\xi }_{{\rm{raw}}}=\arctan (z/D)\), where D = 1.18 m is the distance between the waveguide and detector and z is the vertical position on the detector.

Interference pattern

Two (partially) coherent wave fields E1(x0, z) = A0(z + b/2) and \({E}_{2}({x}_{0},z)={A}_{0}(z-b/2)\sqrt{T}{{\rm{e}}}^{{i}{\phi }}\) with wavelength λ centred at z = ±b/2, where A0(z) describes a complex amplitude with finite width w, produce a far-field Fraunhofer diffraction pattern that can be described using equation (1) (ref. 55). The fringe visibility parameter V0 ∈ [0, 1] accounts for the partial coherence56, for example, due to temporal drifts. The symbol I0(ξ) denotes the Fraunhofer diffraction pattern of A0(z). Setting Teff = 1 + T and \({V}_{{\rm{eff}}}=\sqrt{T}{V}_{0}\) yields

$$I(\xi ;{\bf{\uptheta} })={I}_{0}(\xi ){T}_{{\rm{eff}}}\left[1+{V}_{{\rm{eff}}}\cos (\phi +kb\xi )\right],$$
(4)

with model parameters \({\bf{\uptheta }}=(\phi ,{T}_{{\rm{eff}}},{V}_{{\rm{eff}}})\) and global (constant) parameters: wave number k = 2π/λ, mode width w, waveguide separation b, central angle ξ0 and total intensity Itot. We approximate the exit field of the waveguide to be Gaussian; therefore, its far-field intensity distribution is

$${I}_{0}(\xi )=\frac{{I}_{{\rm{tot}}}}{\sigma \sqrt{\uppi }}\exp [-{(\xi -{\xi }_{0})}^{2}/{\sigma }^{2}],$$
(5)

where σ = 2.355/(kw).

Statistical model

We use a statistical model to implement phase retrieval from the sparse interference patterns. The model parameters \(\bf{\uptheta }\) will be extracted for each detuning Δ independently. The probability to measure yj counts at angle ξj conditional on certain values of \(\bf{\uptheta }\) is \(p({y}_{j}| {\bf{\uptheta} })\)\(={\left.\Pr ({Y}_{j}={y}_{j}| \lambda )\right|}_{\lambda =I({\xi }_{j};{\bf{\uptheta} })}\), where

$$\Pr ({Y}_{j}={y}_{j}| \lambda )=\frac{{\lambda }_{j}^{y}{{\rm{e}}}^{-\lambda }}{{y}_{j}!}$$
(6)

is the Poisson distribution. Since measurements for different angles are independent, we have \(p({\bf{y}}| {\bf{\uptheta} })={\prod }_{j}p({y}_{j}| {\bf{\uptheta} })\). Bayes theorem implies for the posterior distribution:

$$p({\bf{\uptheta} }| {\bf{y}})\propto p({\bf{\uptheta} })\cdot p({\bf{y}}| {\bf{\uptheta} }),$$
(7)

where the proportionality constant only depends on the data \(\bf{y}\) and is independent of \(\bf{\uptheta }\) (ref. 49). We use the ‘emcee’ Markov chain Monte Carlo ensemble sampler50 to compute the posterior, as detailed below.

The Markov chain Monte Carlo sampler is based on the logarithm of the right-hand side of equation (7), which can be computed as follows. The log-likelihood for a given angle ξ is

$$\begin{array}{l}{\ell }_{1}({\bf{\uptheta} };y)\equiv \log [p(y| {\bf{\uptheta} })]\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=-I(\xi ;{\bf{\uptheta} })+y\log [I(\xi ;{\bf{\uptheta} })]-\log [(y!)].\end{array}\,\,\,\,$$
(8)

Therefore, the total log-likelihood is

$$\ell ({\bf{\uptheta} };{\bf{y}})=\mathop{\sum }\limits_{j}\left[-I({\xi }_{j};{\bf{\uptheta} })+{y}_{j}l\mathrm{og}[I({\xi }_{j};{\bf{\uptheta} })]\right]+\,\mathrm{constant}.$$
(9)

We assume uniform prior distributions based on our knowledge of the interference as follows: ϕ ~ U(−π, π), Veff ~ U(0, 1) and Teff ~ U(0.3, 1.3). Although the real part of the exponent in equation (2) can take any value, the interferometer signal is 2π periodic, and hence, the phase is only uniquely retrievable within an interval of length 2π. We slightly loosen the bounds of the transmission in equation (4) from the ideal case.

Phase retrieval

Phase retrieval proceeds as follows.

  1. 1.

    Estimate ξ0, w and Itot from the interference patterns on resonance with Veff = 0 and Teff = 1.

  2. 2.

    Estimate b from a narrow range of off-resonance interference patterns, using the parameters from step 1.

  3. 3.

    Compute a sample ensemble of \(\bf{\uptheta }\) given the data \(\bf{y}\) with emcee, using the global parameters of steps 1 and 2.

  4. 4.

    Estimate the probability density of the phase shift parameter ϕ from a marginal histogram of the ensemble.

  5. 5.

    Shift the phase shift globally such that E[ϕ(+Δ)] = −E[ϕ(−Δ)] for Δ = 20Γ. This eliminates the constant offset (due to the illuminating beam and non-resonant contributions) and leaves the dispersive variation.

Coupling constant

The coupling constant ζm in equation (2) is defined as

$${\zeta }_{{\rm{m}}}=\frac{{[{u}_{{\rm{m}}}({z}_{0})]}^{2}}{{\nu }_{{\rm{m}}}}d,$$
(10)

where νm is the (complex) effective refractive index of the mode, d is the thickness of the iron layer, z0 is the position of the iron layer and um(z) is the transversal profile of the guided mode, which is binormalized to \(\int {[{u}_{{\rm{m}}}(z)]}^{2}/\epsilon (z){\rm{d}}z=1\) (ϵ(z) is the relative electric permeability at height z in the layer system)51. It can be computed numerically, for example, with the xwglib library (ref. 51).

Hyperfine splitting

The 0.6-nm-thick iron layer is amorphous and too thin to develop long-range magnetic order. Thus, it does not exhibit magnetic hyperfine splitting. However, the individual nuclei can generally experience different isomeric shifts and possibly quadrupole splittings, due to their inhomogeneous environments in the film, which is only about two atomic layers thick. We account for this by introducing an effective Gaussian line broadening and a net central shift as

$$\chi (\omega ;{\delta }_{{\rm{HF}}},{w}_{{\rm{HF}}})=f(\omega ;{w}_{{\rm{HF}}})* {\chi }_{0}(\omega -{\delta }_{{\rm{HF}}}),$$
(11)

with

$$f(\omega ;{w}_{\mathrm{HF}})=\frac{1}{{\sigma }_{\mathrm{HF}}\sqrt{2{\rm{\pi }}}}\exp \left\{-\frac{1}{2}{\left(\frac{\omega }{{\sigma }_{\mathrm{HF}}}\right)}^{2}\right\},$$
(12)

where σHF = wHF/2.355.